Is Psycho really the very first film to show a flushing toilet?

According to the Trivia section of Going Wild (1930), Psycho was not the first. However, shortly after 1930 The Hays Code was enacted that prohibited or restricted many things in movies due to the perceived "immorality of Hollywood", including:

1. Pointed profanity—by either title or lip—this includes the words God, Lord, Jesus, Christ (unless they be used reverently in connection with proper religious ceremonies), Hell, S.O.B., damn, Gawd, and every other profane and vulgar expression however it may be spelled;
2. Any licentious or suggestive nudity—in fact or in silhouette; and any lecherous or licentious notice thereof by other characters in the picture;
3. The illegal traffic in drugs;
4. Any inference of sex perversion;
5. White slavery;
6. Miscegenation (sex relationships between the white and black races);
7. Sex hygiene and venereal diseases;
8. Scenes of actual childbirth—in fact or in silhouette;
9. Children's sex organs;
10. Ridicule of the clergy;
11. Willful offense to any nation, race or creed;

and BE IT FURTHER RESOLVED, that special care be exercised in the manner in which the following subjects are treated, to the end that vulgarity and suggestiveness may be eliminated and that good taste may be emphasized:

1. The use of the Flag;

2. International Relations (avoid picturizing in an unfavorable light another country's religion, history, institutions, prominent people and citizenry);

3. Arson;

4. The use of firearms;

5. Theft, robbery, safe-cracking, and dynamiting of trains, mines, buildings, et cetera (having in mind the effect which a too-detailed description of these may have upon the moron);

6. Brutality and possible gruesomeness;

7. Technique of committing murder by whatever method;

8. Methods of smuggling;

9. Third-Degree methods;

10. Actual hangings or electrocutions as legal punishment for crime;

11. Sympathy for criminals;

12. Attitude toward public characters and institutions;

13. Sedition;

14. Apparent cruelty to children and animals;

15. Branding of people or animals;

16. The sale of women, or of a woman selling her virtue;

17. Rape or attempted rape;

18. First-night scenes;

19. Man and woman in bed together;

20. Deliberate seduction of girls;

21. The institution of marriage;

22. Surgical operations;

23. The use of drugs;

24. Titles or scenes having to do with law enforcement or law-enforcing officers;

25. Excessive or lustful kissing, particularly when one character or the other is a "heavy".

This is most likely what led to bathrooms being deemed "inappropriate" and so were not shown in movies. However, again from the above-linked Wiki:

At the forefront of contesting the Code was director Otto Preminger, whose films violated the Code repeatedly in the 1950s. His 1953 film The Moon Is Blue, about a young woman who tries to play two suitors off against each other by claiming that she plans to keep her virginity until marriage, was released without a certificate of approval. He later made The Man with the Golden Arm (1955), which portrayed the prohibited subject of drug abuse, and Anatomy of a Murder (1959), which dealt with murder and rape. Like Some Like It Hot, Preminger's films were direct assaults on the authority of the Production Code, and their success hastened its abandonment. In the early 1960s, films began to deal with adult subjects and sexual matters that had not been seen in Hollywood films since the early 1930s.

By the late 1950s, increasingly explicit films began to appear, such as Anatomy of a Murder (1959), Suddenly, Last Summer (1959), Psycho (1960), and The Dark at the Top of the Stairs (1961). The MPAA reluctantly granted the seal of approval for these films, although not until certain cuts were made. Owing to its themes, Billy Wilder's Some Like It Hot (1959) was not granted a certificate of approval, but it still became a box office smash, and, as a result, it further weakened the authority of the Code.